Two triangles are congruent if they are identical in shape and size — one could be picked up and placed exactly on top of the other. At KS3 you learn four criteria — SSS, SAS, ASA, and RHS — each of which is enough information to guarantee congruence.
What does congruent mean?
Two shapes are congruent if every side and every angle of one matches the corresponding side and angle of the other. They may be in different positions, orientations, or even reflected — but the essential shape is identical.
Congruent triangles have:
- All three pairs of corresponding sides equal in length
- All three pairs of corresponding angles equal in size
You do not need to check all six facts — each of the four criteria below requires only three carefully chosen facts.
What are the four congruence criteria?
| Criterion | Stands for | What you need |
|---|---|---|
| SSS | Side-Side-Side | All three sides equal |
| SAS | Side-Angle-Side | Two sides and the included angle equal |
| ASA | Angle-Side-Angle | Two angles and the included side equal |
| RHS | Right angle-Hypotenuse-Side | A right angle, the hypotenuse, and one other side equal |
The word "included" is key: in SAS, the angle must be the one between the two given sides. In ASA, the side must be the one between the two given angles. Getting this wrong is the most common error.
Criterion 1: SSS (Side-Side-Side)
If all three sides of one triangle equal the corresponding three sides of another, the triangles are congruent.
Example: Triangle ABC has sides 5 cm, 7 cm, 9 cm. Triangle PQR has sides 9 cm, 5 cm, 7 cm.
Even though the sides are listed in a different order, the same three lengths exist in both triangles. Match: AB = PQ = 5 cm (say), BC = QR = 7 cm, AC = PR = 9 cm.
Conclusion: Congruent by SSS.
Criterion 2: SAS (Side-Angle-Side)
If two sides and the angle between them (the included angle) are equal, the triangles are congruent.
Example: In triangle DEF, DE = 6 cm, EF = 4 cm, and angle DEF = 50°. In triangle GHI, GH = 6 cm, HI = 4 cm, and angle GHI = 50°.
The angle E is between sides DE and EF. The angle H is between sides GH and HI. Both are 50° and both pairs of enclosing sides are equal.
Conclusion: Congruent by SAS.
Common trap: If the equal angle is not between the two known sides, SAS does not apply — it becomes the non-criterion SSA, which does not guarantee congruence.
Criterion 3: ASA (Angle-Side-Angle)
If two angles and the side between them (the included side) are equal, the triangles are congruent.
Example: Triangle JKL has angle J = 40°, JK = 8 cm, angle K = 65°. Triangle MNO has angle M = 40°, MN = 8 cm, angle N = 65°.
Side JK lies between angles J and K. Side MN lies between angles M and N. Both sides are 8 cm and the surrounding angles match.
Conclusion: Congruent by ASA.
Note: if two angles are equal, the third angle must also be equal (since angles in a triangle sum to 180°). So ASA is equivalent to AAS (Angle-Angle-Side) provided you are careful about which side is given.
Criterion 4: RHS (Right angle-Hypotenuse-Side)
This criterion applies only to right-angled triangles. If both triangles have a right angle, equal hypotenuses, and one other pair of equal sides, they are congruent.
Example: Triangle PQR has a right angle at Q, hypotenuse PR = 10 cm, and leg QR = 6 cm. Triangle STU has a right angle at T, hypotenuse SU = 10 cm, and leg TU = 6 cm.
Conclusion: Congruent by RHS.
The hypotenuse is always the side opposite the right angle — the longest side. The third side (the remaining leg) would be √(10² − 6²) = √64 = 8 cm in both triangles by Pythagoras, which explains why two of the three sides plus the right angle are sufficient.
Why isn't SSA (Side-Side-Angle) a valid criterion?
SSA — two sides and a non-included angle — is not a reliable criterion because two different triangles can share the same SSA combination. This is called the ambiguous case.
Imagine you draw side a = 5 cm from a point, mark a 30° angle at one end, and try to position side b = 4 cm to close the triangle. Depending on the exact lengths, the far end of the 4 cm side may reach the opposite side in two different positions, giving two non-congruent triangles — both with the same SSA data.
Frequently asked questions
How do I decide which criterion to use?
Work through a checklist: count how many sides and angles you know are equal.
- Three sides equal → SSS
- Two sides equal and the angle between them → SAS
- Two angles equal and the side between them → ASA
- Right angle + hypotenuse + one side → RHS
If you have two angles, you automatically know all three (they add to 180°), so AAS becomes ASA with a bit of thought.
Do I need to name corresponding vertices in the right order?
Yes. When stating that triangles are congruent you should write the vertices in matching order: "Triangle ABC ≅ Triangle DEF" means A corresponds to D, B to E, and C to F. Getting the correspondence wrong earns no marks for the conclusion, even if the criterion is identified correctly.
What is the difference between congruent and similar?
Congruent triangles are identical in both shape and size. Similar triangles have the same shape (all angles equal) but may be different sizes (sides in proportion, not necessarily equal). Two congruent triangles are automatically similar; two similar triangles are congruent only if their scale factor is 1.
Does SSS always produce a unique triangle?
Yes — given three specific side lengths that satisfy the triangle inequality (each side shorter than the sum of the other two), there is exactly one triangle (up to reflection and rotation). This is the geometric reason SSS guarantees congruence.
Explore congruence criteria with Professor Pi's guided questions at aitutors.me.